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"It is impossible for a man to
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 Epictete
Math
Shortcuts
The
radii of circumscribed (R)
and inscribed (r) circles
within regular polygons of n sides, each of
length x, are given by:
(x/2) csc(180°/n)
and
(x/2) cot(180°/n)


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Puzzles of the Month + Solutions



Back to PuzzleoftheMonth
page  Home 
Puzzle
# 125


A
mathematical shield
Once
upon a time, Mars, the God of war, intended to test the IQ of the goddess Minerva.
So, showing his shield, he told her: "Darling, on my shield, there are 3
equal circles which represent the qualities of the warrior: strength, flexibility,
and decisiveness. As you can see, one of the circles has been scratched by a
sword, and the resulting score is three inches long (line AB in the illustration).
Can you then tell me what is the area of my shield?".
Find the very shortest way to solve
this puzzle and use only basic geometry, trigonometry
is not allowed!
Difficulty
level: ,
basic geometry knowledge.
Category:
Geometrical puzzle.
Keywords:
inscribed circles, incircles.
Related
puzzles:
 Red monad,
 Achtung Minen!
 Lo
scudo matematico
Un giorno, il dio della guerra Marte volle mettere
alla prova l'intelligenza della dea Minerva e, mostrando
il suo scudo, gli disse: "Carissima, sul mio
scudo ci sono 3 cerchi uguali che rappresentano le
qualità del guerriero: concentrazione, flessibilità,
risolutezza. Come puoi vedere, uno dei cerchi è stato
scalfito da una spada, la riga che ne consegue (linea
AB, vedi l'illustrazione) è lunga tre pollici.
Sapresti dirmi qual è l’area del mio
scudo?".
Grazie di non utilizzare la trigonometria nella risoluzione
del problema!
Parole
chiave: cerchi inscritti.
Suggerisci un'altra
soluzione Chiudi
 Un
bouclier mathématique
Un jour, Mars le dieu de la guerre voulut tester
l'intelligence de la déesse Minerve et, montrant
son bouclier, lui dit: "Ma chère, sur
mon bouclier, se trouvent 3 cercles égaux
qui représentent les qualités du guerrier, à savoir:
force, souplesse, détermination. Comme tu
peux le remarquer, l'un des cercles a été éraflé par
une épée, la rayure qui en suit (ligne
AB, voir illustration) est longue de trois pouces.
Pourraistu me dire quelle est la superficie de mon
bouclier?".
Merci
de ne pas utiliser la trigo pour résoudre
ce puzzle!
Mots
clés: cercles inscrits.
Propose une
autre solution Fermer

Source
of the puzzle:
© G. Sarcone. You
cannot reproduce any part of this page without prior written
permission. 



The
centers of the small green circles are equidistant
from each other, and thus form the vertices of an
equilateral triangle with midpoints A, B and C (see
image below).
It follows that the small triangles ABC and AC'B are
also equilateral and identical to each other. The radii r of
the small circles are congruent to the sides of the
triangles ABC and AC'B, then
r = C'B = AB.
As
shown in the image, the
radius R of the large circle can be calculated
by adding r + h + NM together.
We
can calculate the height h of
the triangle AC'B by multiplying its side AB by √3/2:
h = (AB√3)/2 = (3√3)/2
Since
the heights of an equilateral triangle meet at a
point (here, point M) that is two thirds
of the distance from the vertex of the triangle to
the base, we can also calculate MN with
this simple formula:
MN = h x 1/3 = (3√3)/2 x 1/3
= (3√3)/2 x 3 = √3/2
Therefore,
the area of large yellow circle (which represents
the shield) is:
π[3
+ (3√3)/2 + √3)/2]^{2} = π(3
+ 2√3)^{2} ≈ 131.27 square
inches

The
5 Winners of the Puzzle of the Month are:
John Pelot, USA  Benoît
Humbert, France  Amedeo
Squeglia, Italy  Paritosh
Singh, India  Walter
Jacobs, Belgium
Congratulations!



Math
fact behind the puzzle

Properties
of the equilateral triangle
An equilateral triangle is simply a specific case of
a regular polygon, in this case with 3 sides.
Equilateral triangles are triangles in which all sides
are equal, and all angles are equal as well and each
of them measures 60Å.
With an equilateral triangle, the radius of the incircle
is exactly half the radius of the circumcircle. 
© 2006 G.
Sarcone, www.archimedeslab.org
You can reuse content from Archimedes’ Lab
on the ONLY condition that you provide credit to the
authors (© G.
Sarcone and/or M.J.
Waeber) and a link back to our site. You CANNOT
reproduce the content of this page for commercial
purposes.

You're
encouraged to expand and/or improve this article. Send
your comments, feedback or suggestions to Gianni
A. Sarcone. Thanks! 




